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Theorem f1omo 47527
Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 47526 assuming ax-un 7725 (see f1omoALT 47528). (Contributed by Zhi Wang, 19-Sep-2024.)
Hypothesis
Ref Expression
f1omo.1 (𝜑𝐹 = (𝐴 × {1o}))
Assertion
Ref Expression
f1omo (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1oex 8476 . . . 4 1o ∈ V
2 eqid 2733 . . . 4 ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
31, 2fvconst0ci 47525 . . 3 (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4 mo0 47498 . . . 4 (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5 el1o 8495 . . . . . . . 8 (𝑦 ∈ 1o𝑦 = ∅)
6 el1o 8495 . . . . . . . 8 (𝑥 ∈ 1o𝑥 = ∅)
7 eqtr3 2759 . . . . . . . 8 ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥)
85, 6, 7syl2anb 599 . . . . . . 7 ((𝑦 ∈ 1o𝑥 ∈ 1o) → 𝑦 = 𝑥)
98gen2 1799 . . . . . 6 𝑦𝑥((𝑦 ∈ 1o𝑥 ∈ 1o) → 𝑦 = 𝑥)
10 eleq1w 2817 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ∈ 1o𝑥 ∈ 1o))
1110mo4 2561 . . . . . 6 (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦𝑥((𝑦 ∈ 1o𝑥 ∈ 1o) → 𝑦 = 𝑥))
129, 11mpbir 230 . . . . 5 ∃*𝑦 𝑦 ∈ 1o
13 eleq2w2 2729 . . . . . 6 (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o))
1413mobidv 2544 . . . . 5 (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o))
1512, 14mpbiri 258 . . . 4 (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
164, 15jaoi 856 . . 3 ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
173, 16ax-mp 5 . 2 ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
18 f1omo.1 . . . . 5 (𝜑𝐹 = (𝐴 × {1o}))
1918fveq1d 6894 . . . 4 (𝜑 → (𝐹𝑋) = ((𝐴 × {1o})‘𝑋))
2019eleq2d 2820 . . 3 (𝜑 → (𝑦 ∈ (𝐹𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
2120mobidv 2544 . 2 (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
2217, 21mpbiri 258 1 (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2533  c0 4323  {csn 4629   × cxp 5675  cfv 6544  1oc1o 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-1o 8466
This theorem is referenced by:  indthinc  47672  prsthinc  47674
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